- 00-195 G.M. Molchan
- Maximum of fractional Brownian motion: probabilities of small values
(43K, LATeX)
Apr 21, 00
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Abstract. Let $b_\gamma (t)$, $b_\gamma(0)=0$ be fractional Brownian
motion, i.e., a Gaussian process with the structure
function $E|b_\gamma (t) - b_\gamma (s)|^2 = |t-s|^\gamma$, $0<\gamma<2$.
We study the logarithmic asymptotics of
$P_T = P\{b_\gamma (t) < 1,\quad t \in T\Delta \}$ as
$T \to \infty$, where $\Delta$ is either the interval $(0,1)$
or a bounded region that contains a vicinity of $0$ for the case of
multidimensional time. It is shown that
$\log\,P_T = -D\log\,T(1+o(1))$, where $D$ is the dimension of zeroes
of $b_\gamma(t)$ in the former case and the dimension of time in
the latter.
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